Question: Simplify the following expression and state the conditions under which the simplification is valid. You can assume that $p \neq 0$. $t = \dfrac{p^2 + 2p - 35}{p + 3} \times \dfrac{p + 3}{-7p + 35} $
Solution: First factor the quadratic. $t = \dfrac{(p - 5)(p + 7)}{p + 3} \times \dfrac{p + 3}{-7p + 35} $ Then factor out any other terms. $t = \dfrac{(p - 5)(p + 7)}{p + 3} \times \dfrac{p + 3}{-7(p - 5)} $ Then multiply the two numerators and multiply the two denominators. $t = \dfrac{ (p - 5)(p + 7) \times (p + 3) } { (p + 3) \times -7(p - 5) } $ $t = \dfrac{ (p - 5)(p + 7)(p + 3)}{ -7(p + 3)(p - 5)} $ Notice that $(p + 3)$ and $(p - 5)$ appear in both the numerator and denominator so we can cancel them. $t = \dfrac{ \cancel{(p - 5)}(p + 7)(p + 3)}{ -7(p + 3)\cancel{(p - 5)}} $ We are dividing by $p - 5$ , so $p - 5 \neq 0$ Therefore, $p \neq 5$ $t = \dfrac{ \cancel{(p - 5)}(p + 7)\cancel{(p + 3)}}{ -7\cancel{(p + 3)}\cancel{(p - 5)}} $ We are dividing by $p + 3$ , so $p + 3 \neq 0$ Therefore, $p \neq -3$ $t = \dfrac{p + 7}{-7} $ $t = \dfrac{-(p + 7)}{7} ; \space p \neq 5 ; \space p \neq -3 $